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Theorem islly 21265
Description: The property of being a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
islly (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐽,𝑥,𝑦

Proof of Theorem islly
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3805 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
2 oveq1 6654 . . . . . . 7 (𝑗 = 𝐽 → (𝑗t 𝑢) = (𝐽t 𝑢))
32eleq1d 2685 . . . . . 6 (𝑗 = 𝐽 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝐽t 𝑢) ∈ 𝐴))
43anbi2d 740 . . . . 5 (𝑗 = 𝐽 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
51, 4rexeqbidv 3151 . . . 4 (𝑗 = 𝐽 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
65ralbidv 2985 . . 3 (𝑗 = 𝐽 → (∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∀𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
76raleqbi1dv 3144 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
8 df-lly 21263 . 2 Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
97, 8elrab2 3364 1 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  cin 3571  𝒫 cpw 4156  (class class class)co 6647  t crest 16075  Topctop 20692  Locally clly 21261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-lly 21263
This theorem is referenced by:  llytop  21269  llyi  21271  llyss  21276  subislly  21278  restnlly  21279  restlly  21280  islly2  21281  llyrest  21282  llyidm  21285  dislly  21294  txlly  21433  ismntop  30055  cnllysconn  31212  rellysconn  31218
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